WIMP Dark Matter through the Dilaton Portal
WWIMP Dark Matter through theDilaton Portal
Kfir Blum a , Mathieu Cliche b , Csaba Cs´aki b , and Seung J. Lee c,d a Institute for Advanced Study, Princeton, nj usa b Department of Physics, lepp , Cornell University, Ithaca, ny usa c Department of Physics, Korea Advanced Institute of Science and Technology, 335Gwahak-ro, Yuseong-gu, Daejeon 305-701,
Korea d School of Physics, Korea Institute for Advanced Study, Seoul 130-722,
Korea
Abstract
We study a model in which dark matter couples to the Standard Model through adilaton of a sector with spontaneously broken approximate scale invariance. Scaleinvariance fixes the dilaton couplings to the Standard Model and dark matter fields,leaving three main free parameters: the symmetry breaking scale f , the dilaton mass m σ , and the dark matter mass m χ . We analyze the experimental constraints on theparameter space from collider, direct and indirect detection experiments includingthe effect of Sommerfeld enhancement, and show that dilaton exchange provides aconsistent, calculable framework for cold dark matter with f, m σ , m χ of roughlysimilar magnitude and in the range ∼ −
10 TeV. Direct and indirect detectionexperiments, notably future ground-based gamma ray and space-based cosmic raymeasurements, can probe the model all the way to dark matter mass in the multi-TeV regime.
Embedding the Standard Model (SM) partially or completely in a composite sector cansolve the hierarchy problem, by making the Higgs boson composite. Often such a compositesector arises as the low-energy limit of an approximately scale invariant theory, where scaleinvariance is broken somewhere above the weak scale. If the breaking of scale invariance isspontaneous, then it is accompanied by a light dilaton σ that couples to the fields in thecomposite sector through the trace of the energy-momentum tensor [1–6] − σf Tr T. (1.1)For massive particles, the coupling to σ is proportional to the particle masses, with thesuppression scale f corresponding to the breaking of scale invariance.The canonical dilaton Lagrangian (1.1) offers an economical way to couple the SM tonew fields that could be singlets under the SM gauge symmetries and thus form an otherwise a r X i v : . [ h e p - ph ] O c t ark sector. In this paper we study the possibility that dark matter (DM) belongs to suchdark sector and couples to the SM through Eq. (1.1). In the minimal set up that we explorehere, three parameters determine the dynamics of thermal freeze-out in the early Universe:the breaking scale f , the dilaton mass m σ , and the dark matter mass m χ . Fixing one ofthese parameters such that the observed dark matter relic abundance is reproduced leavesa rather predictive framework. We show that a large parametric region exists where thesolution is perturbative and produces cold, weakly interacting massive particle dark matter(WIMP), with f, m σ , m χ of roughly similar magnitude and in the range ∼ −
10 TeV.Null results from dark matter direct detection experiments like LUX [7], XENON100 [8]and CDMS [9] put considerable pressure on WIMP models where DM couples to the SMthrough exchange of SM particles. The annihilation cross section σ ann ∼ − cm , requiredfor WIMP relic abundance consistent with observations, is some ten orders of magnitudelarger than the WIMP-nucleon elastic scattering cross section now probed by the directdetection experiments. This excludes Z boson exchange in all but fine-tuned corners ofthe parameter space, and requires some tuning for Higgs mediation as well. In contrast,the dilaton portal we analyze here quite generically evades the direct detection constraintsin the bulk of the relevant parameter space, as the DM coupling to the SM resemblesthe case of Higgs exchange but with extra suppression of order ( v/f ) ( m h /m σ ) with v = (cid:104) H (cid:105) = 246 GeV and m σ and the scale f automatically lying in the TeV ballpark toprovide the correct relic abundance.The idea that dark matter could couple to the SM via dilaton exchange was analyzedpreviously in Ref. [10] (where the dilaton was taken to be massless) and in Ref. [11] (forsome specific warped extra dimensional models where the role of the dilaton was played bythe radion). Our work generalizes the results of Ref. [11] and extends the analysis of [10]by adding the dilaton mass as a free parameter. This allows a more complete explorationof the parameter space and reveals effects such as Sommerfeld-enhanced annihilation. Wealso incorporate the most recent experimental bounds from direct and indirect detectionas well as collider experiments.The outline of this paper is as follows. In Sec. 2 we summarize the basic propertiesof the dilaton. We present its couplings, fixed mainly by the scale f with a few additionalparameters characterizing the embedding of the SM matter into the composite sector,comment on expected NDA bounds on the dilaton mass, and present two benchmark modelsto be studied in the paper. Sec. 3 contains a calculation of the DM annihliation cross sectiondue to dilaton exchange. After deriving a unitarity bound on the DM mass, we presentthe parameter space of the theory where the observed relic abundance is reproduced. InSec. 4 we compare the DM-nucleon scattering cross sections to the experimental boundsfrom the latest round of direct detection measurements, finding that large regions of theparameter space are compatible with the bounds. In Sec. 5 we consider constraints fromindirect detection of gamma rays and cosmic ray antiprotons. We conclude in Sec. 6. App.A summarizes collider bounds on the dilaton, considering LEP, Tevatron and the LHC.App. B contains cross-section formulae for the sub-leading annihilation channels that we2mit in the body of the text for clarity. We start by considering the effective theory describing an approximately scale invariantsector with scale invariance spontaneously broken at the scale f . The Goldstone bosoncorresponding to this breaking, called the dilaton σ ( x ), can be parametrized via a spurionfield Φ( x ) as [12] Φ( x ) ≡ f e σ ( x ) /f (2.1)such that under a scale transformation x → xe λ we have Φ( x ) → e λ Φ( e λ x ) and (cid:104) Φ (cid:105) = f . Toobtain a canonically normalized dilaton kinetic term it is convenient to do a field redefinitionsuch that Φ( x ) = σ ( x )+ f [5] where σ is now the canonically normalized dilaton field. Usinga spurion analysis one can then find the low energy theory below the cutoff scale 4 πf byinserting powers of Φ /f in the SM Lagrangian to make it scale invariant. After electroweaksymmetry breaking one finds the following effective action describing the interactions ofthe canonically normalized dilaton with the SM fields [4, 5, 12] L σ = 12 ∂ µ σ∂ µ σ − m σ σ − m σ f σ − m σ f σ + . . . − (cid:18) σf (cid:19) (cid:34)(cid:88) ψ (1 + γ ψ ) m ψ ¯ ψψ (cid:35) ++ (cid:18) σf + σ f (cid:19) (cid:20) m W W + µ W − µ + 12 m Z Z µ Z µ − m h h (cid:21) + α EM πf c EM σF µν F µν ++ α s πf c G σG aµν G aµν . (2.2)The sum on ψ runs over the SM fermions, which are assumed to be partially compositewith light fermions being mainly elementary and the top quark mainly composite. γ ψ corresponds to the anomalous dimension of fermionic operators responsible for generatingthe SM fermion masses after mixing between the elementary and composite sectors. Forcomposite fermions the anomalous dimension is expected to be small γ ψ (cid:39)
0, while for lightfermions the anomalous dimension may be sizable.Naive dimensional analysis (NDA) limits the plausible size of the dilaton mass. Forexample, considering the dilaton self-energy loop from the trilinear coupling in Eq. (2.2)we find that m σ ≤ πf (2.3)to ensure that the one-loop correction of the dilaton mass remains below the tree-levelmass, and that the couplings of the dilaton to matter remain under control [13]. This isjust the reflection of the fact that this theory has an intrinsic cutoff of order Λ ∼ πf , andwe should treat it as an effective theory valid below that scale. Note also that it is difficultto make the dilaton much lighter than the cutoff scale. In generic models there is a tuning3f order m σ Λ necessary to lower the dilaton mass [12–14], though special constructions canpotentially alleviate this tuning [15, 16]. We will require that the dilaton is not lighter than f / , and small explicit sources of scale symmetry breaking such asΦ − (cid:15) . Requiring that (cid:104) Φ (cid:105) = f and that d V (Φ) d Φ = m σ fixes the parameters of the dilatonpotential. The leading expression for the cubic self-coupling of the dilaton is 5 / (cid:15) → /
24. Away from the (cid:15) → / , /
6] [5]. For simplicity, throughout this paper we have usedthe limiting value 5 / W boson loops, but in addition there is a direct contribution from the trace anomaly.The trace anomaly is proportional to the β -functions: the actual contribution will be thedifference between the β -function above and below the symmetry breaking scale. Thus thiscontribution depends on the details of what fraction of the composite sector is actuallycharged under the unbroken SM gauge symmetries, and what fraction of the SM fields arecomposites. For example, the coupling to gluons c G receives a contribution from the traceanomaly and from a top loop and is given by c G = b (3)IR − b (3)UV + 12 F / ( x t ) (2.4)where b (3) UV,IR are the QCD β -function coefficients above and below the scale f . This is a freeparameter of the theory, which gives a measure of the QCD charges of the scale invariantsector. The function F / is the usual triangle diagram contribution of a fermion given by F / ( x ) = 2 x [1 + (1 − x ) f ( x )] (2.5) f ( x ) = (cid:2) sin − (1 / √ x ) (cid:3) x ≥ − (cid:104) log (cid:16) √ x − −√ x − (cid:17) − iπ (cid:105) x < x t = 4 m t /m σ [4, 12]. A similar expression applies to the coupling to photons.Some of the results in the following sections (in particular the direct detection and col-lider signals) depend on the parameters c G and c EM . To this end we define two benchmarkmodel examples which we will study in detail. Model A:
This is the well-studied case proposed in [5] where the entire SM is composite,corresponding to b UV = 0 , b IR = b SM , giving rise to the parameters b UV − b IR = − , b EMUV − b EMIR = 11 /
3. Note that for a light dilaton these b ’s depend somewhat onthe dilaton mass: for example b UV − b IR = −
11 + 2 n/
3, with n denoting the numberof quarks whose mass is smaller than m σ / odel B: This is a limit of the well-motivated case when only the right-handed top andthe Goldstone bosons needed for electroweak symmetry breaking are composites,while we minimize the β -functions of the UV to be as small as possible, resulting in b UV = b EMUV = 0 , b IR = − / , b EMIR = − /
9. Note however that b UV is in fact a freeparameter depending on the actual UV theory, and its value here has been chosenonly for illustration.The final ingredient of the model is χ , the dark matter particle, which can be spin 0,1/2 or 1. We assume that χ is a composite of the conformal sector, and does not have anydirect coupling to the standard model fields which are mainly elementary. The couplings of χ to the dilaton are fixed by a spurion analysis and follow the rules of couplings of genericmassive composites [10]: L DM ⊃ − (cid:16) σf + σ f (cid:17) m χ χ Scalar − (cid:16) σf (cid:17) m χ ¯ χχ Fermion (cid:16) σf + σ f (cid:17) m χ χ µ χ µ Gauge boson. (2.7)For simplicity we assume that a Z symmetry renders χ to be a stable particle. For thefermionic case, we assume that χ is a Dirac fermion. In this section we present the calculation of the relic abundance of the dark matter field χ , where annihilations into SM states are assumed to proceed via dilaton exchange, andexhibit the relevant parameter space of the theory. As usual, for small relative velocities v the velocity-weighted annihilation cross section can be expanded as σv = a + bv . At thefreeze-out temperature T F we have (cid:104) v (cid:105) = 6 /x F where x F = m χ /T F . The value of x F canthen be determined by solving the Boltzmann equation in an expanding Universe: x F = ln (cid:32) (cid:114) g π M Pl m χ ( a + 6 b/x F ) √ g ∗ √ x F (cid:33) , (3.1)where g is the number of degrees of freedom of the dark matter particle and g ∗ is the effectivenumber of relativistic degrees of freedom in thermal equilibrium during dark matter freeze-out. Once x F is determined the dark matter relic abundance is given byΩ χ h ≈ . × GeV M Pl √ g ∗ x F a + 3( b − a/ /x F . (3.2)As we show below, the dark matter annihilation cross section (and thus the parameters a, b ) in the model considered here is calculated in terms of m χ , m σ and f . Requiring that5igure 1: Leading annihilation diagrams of dark matter in the regime m χ (cid:29) m t . Forfermionic dark matter there is no direct annihilation to dilatons.the observed relic abundance Ω χ h = 0 . ± . The dominant dark matter annihilation channels for m χ (cid:29) m t are χχ → σσ, W W, ZZ ,shown in Fig. 1. The dominant channels contain factors of m χ /f , to be compared with allother sub-leading channels (for example s-channel dilaton exchange with quark or higgs finalstates) that contain factors of m q /f or m h /f instead and are thus suppressed by relativepowers of m q,h /m χ . Below we present analytical expressions for the dominant channels inthe limit m χ (cid:29) m t , for the cases of scalar, fermion and vector dark matter. Formulae forthe sub-leading annihilation channels can be found in Appendix B. For numerical resultsall of the allowed annihilation channels are included. Scalar dark matter
Scalar dark matter annihilation is dominated by s-wave processes. The approximate ex-pressions of the cross sections are σv ( χχ → W W ) (cid:39) m χ m W (cid:113) m χ − m W (cid:16) (2 m χ − m W ) m W (cid:17) πf | m χ − m σ − i Im (cid:0) Π(4 m χ ) (cid:1) | , (3.3) σv ( χχ → σσ ) (cid:39) m χ (cid:112) m χ − m σ | m χ − m σ ) + 2 m σ − i Im (cid:0) Π(4 m χ ) (cid:1) (cid:0) m χ + m σ (cid:1) | πf (2 m χ − m σ ) | m χ − m σ − i Im (cid:0) Π(4 m χ ) (cid:1) | . (3.4)6ote that the second term in the parenthesis of Eq. (3.3), corresponding to the formation oflongitudinal gauge boson modes, becomes proportional to m χ /m W in the limit m χ (cid:29) m W .In this limit, the m W pre-factor is cancelled such that the overall cross section scales like m χ /f .In the expressions above Π( p ) is the 1PI self-energy insertion for the dilaton, whichon-shell is related to the width via m σ Γ σ = − Im (Π( m σ )). Note that we only include theimaginary part in our calculations. The real part (once properly renormalized) is expectedto be a moderate correction to the existing real part of the propagator, which will result insmall shifts to the precise shape of the contours presented below, but can not qualitativelychange the results, as long as the NDA bound (2.3) on the dilaton mass is obeyed. On theother hand properly incorporating the non-vanishing imaginary part can give significantshifts in the resulting cross sections especially close to the resonance.The total width of the dilaton is the sum of the partial widths to Higgs, quarks, massivegauge bosons and dark matter, which in the limit m σ (cid:29) m t is dominated by the decays tomassive gauge bosonsΓ σ ( σ → W W ) = m W πm σ f (cid:115) − m W m σ (cid:18) m σ − m W ) m W (cid:19) . (3.5)The processes χχ → ZZ and σ → ZZ are obtained from Eqs. (3.3-3.5) by replacing m W by m Z and dividing by 2 to account for the phase space of identical final state particles. InAppendix C we collect the contributions of the other channels to the dilaton decay width. Fermionic dark matter
For fermionic dark matter, the annihilation channels have no s-wave contribution, thus thedominant contribution is a p-wave process which is suppressed by a factor of v . We find σv ( χ ¯ χ → W W ) (cid:39) v m χ m W (cid:113) m χ − m W (cid:16) (2 m χ − m W ) m W (cid:17) πf | m χ − m σ − i Im (cid:0) Π(4 m χ ) (cid:1) | (3.6) σv ( χ ¯ χ → σσ ) (cid:39) v (cid:34) m χ (cid:112) m χ − m σ (cid:0) m χ − m σ m χ + 2 m σ (cid:1) πf (cid:0) m χ − m χ m σ + 24 m χ m σ − m σ m χ + m σ (cid:1) + 25 m χ m σ (cid:112) m χ − m σ πf | m χ − m σ − i Im (cid:0) Π(4 m χ ) (cid:1) | − m χ m σ (cid:112) m χ − m σ (5 m χ − m σ )48 πf (cid:0) m χ − m σ m χ + m σ (cid:1) Re (cid:32) m χ − m σ − i Im (cid:0) Π(4 m χ ) (cid:1) (cid:33) (cid:35) . (3.7)7 ector dark matter For vector boson dark matter the annihilation is again dominated by s-wave processes: σv ( χχ → W W ) (cid:39) m χ m W (cid:113) m χ − m W (cid:16) (2 m χ − m W ) m W (cid:17) πf | m χ − m σ − i Im (cid:0) Π(4 m χ ) (cid:1) | (3.8) σv ( χχ → σσ ) (cid:39) m χ (cid:112) m χ − m σ πf (2 m χ − m σ ) | m χ − m σ − i Im (cid:0) Π(4 m χ ) (cid:1) | (cid:16) m χ + 44 m σ m χ Im (cid:0) Π(4 m χ ) (cid:1) − m σ Im (cid:0) Π(4 m χ ) (cid:1) − m σ m χ + 1424 m χ m σ − m σ m χ + 11 m σ Im (cid:0) Π(4 m χ ) (cid:1) + 96 m σ (cid:17) . (3.9) We emphasize again that the
W W and ZZ annihilation channels are important because ofthe enhanced contributions of the longitudinal modes. Note that Ref. [10] neglected thesechannels due to the suppression of the W/Z couplings by m W,Z /f . However as we haveshown in the previous section, these factors are cancelled in the limit m χ (cid:29) m Z due tothe contributions of the longitudinal modes which grow with the CM energy/dark mattermass.For large DM mass, the gauge boson longitudinal modes might violate unitarity. Thisis analogous to the unitarity violation in elastic WW scattering in the standard modelwithout the Higgs. However here the Higgs does not save unitarity. Thus we will have aunitarity bound on the DM mass, related to the built-in cutoff for the theory above whichit is expected to be strongly coupled. One can estimate the unitarity bound on m χ byconsidering the contribution of the longitudinal mode to the scattering amplitude in thelarge DM mass limit, given by M ≈ m χ /f for either scalar, fermion or vector DM. Theresulting s-wave partial wave amplitude a ≈ m χ / (16 πf ) satisfies the unitarity bound |(cid:60) ( a ) | ≤ / m χ ≤ √ πf. (3.10)This unitarity bound on m χ is slightly more constraining than the NDA estimate for thecutoff m χ (cid:46) Λ NDA = 4 πf . A similar analysis for the annihilation to dilatons results in thesame upper bound. We now analyze the parameter space of the model that is compatible with the observeddark matter relic density. Fig. 2 shows the available parameter space where the observedrelic density can be reproduced by an appropriate choice of the symmetry breaking scale f .8he top left, top right, and bottom panels show the results for scalar, fermion, and vectordark matter, respectively. The x - and y -axes correspond to the dilaton and dark mattermass, while the contours show the value of f that is required to obtain the observed darkmatter relic density.For concreteness, in the rest of this section we discuss the scalar dark matter case.We later summarize the results for fermion and vector dark matter. To understand theresults shown in Fig. 2 (top left, as we are focusing on the scalar example), we considerthe different parametric regions in turn. Consider the case m χ , m σ (cid:29) m Z , where annihi-lation to W W, ZZ and, if kinematically allowed, σσ dominates. Assume first m χ > m σ ,corresponding to the upper-left region in Fig. 2. Here we have (cid:104) σv (cid:105) ≈ m χ πf ≈ × − (cid:18) f (cid:19) − (cid:18) m χ f (cid:19) cm / s (valid for m χ (cid:29) m σ ) . (3.11)Recall that relic abundance consistent with observations requires (cid:104) σv (cid:105) ≈ × − cm /s,and that Ω χ h ∝ (cid:104) σv (cid:105) − , imposing the relation m χ = f / (6TeV). Combining this withthe unitarity bound m χ ∼ √ πf obtained above, we find an upper bound f <
30 TeV.Violating this bound leads to DM annihilation cross section that is too small, and so DMrelic density that is too high to match observations. A caveat in this derivation is thatour dark matter particle may co-annihilate with extra particles in the dark sector. If thisco-annihilation is efficient, due to some mass degeneracy in the dark sector and large crosssections, then it would relax the bound on f , allowing f to be somewhat larger than 30 TeV.Even taking this caveat into account, a rough bound f (cid:46)
100 TeV is still expected to hold.We note that this derivation of the bound on f is compatible with the unitarity argumentof [18], that showed that m χ (cid:46)
100 TeV is required in general from S-matrix unitarity (weupdate their early result here by using the currently measured DM relic density). Pluggingthe model-independent upper bound on m χ from Ref. [18] into Eq. (3.11), we obtain again f (cid:46)
30 TeV. The consistency between Eq. (3.10) and the unitarity bound of [18] impliesthat Eq. (3.10) is satisfied throughout the parameter space shown in Fig. 2.Next, consider the region with m σ (cid:29) m χ , so that the χχ → σσ channel is kinematicallyforbidden. This region corresponds to the lower-right part of Fig. 2. In this regime, andstill assuming m χ (cid:29) m Z , one finds the following approximate form for the cross section: (cid:104) σv (cid:105) ∼ m χ πf m σ ≈ · − (cid:16) m χ
350 GeV (cid:17) (cid:18) TeV f (cid:19) (cid:18) TeV m σ (cid:19) cm / s ( m χ (cid:29) m W , m σ (cid:29) m χ ) . (3.12)As one increases the dilaton mass m σ the symmetry breaking scale f needs to decrease inorder to keep the relic abundance fixed. However, one will very quickly need to lower f below the value m σ / π , implying that we have left the regime of validity of our effectivetheory. Therefore most of the lower left region will be excluded based on this criterion. Ofcourse the exact shape of the excluded region will be somewhat uncertain: it depends on theexact onset of strong coupling, and can also be slightly modified by strong co-annihilations9 .1 2.42.73 33.3 3.3 3.63.63.9 4.24.22.0 2.5 3.0 3.5 4.0 4.5 5.02.02.53.03.54.04.55.0 Log [ m σ / GeV ] Log [ m χ / G e V ] Log [ f / GeV ] such that ρ = ρ ( scalar DM ) [ m σ / GeV ] Log [ m χ / G e V ] Log [ f / GeV ] such that ρ = ρ ( fermion DM ) [ m σ / GeV ] Log [ m χ / G e V ] Log [ f / GeV ] such that ρ = ρ ( vector DM ) Figure 2: Parameter space for scalar (top left), fermion (top right) and vector (bottom)dark matter with freeze-out mediated by dilaton exchange. The x and y axes correspond tothe dilaton and dark matter mass, respectively. Contours show the value of the symmetrybreaking scale f , that is required in order to obtain the observed dark matter relic density.In the blank region in the lower-right part of the plot, there is no real solution for f that provides the correct relic density while satisfying Eq. (2.3). Above the red dashedline m σ < f /
10, signaling some degree of fine-tuning. Note that the model-independentunitarity bound of Ref. [18] implies m χ (cid:46) GeV (see text).10n the dark sector. Nevertheless, even in this case we expect that the allowed region wouldshift only slightly.The resonance at m σ = 2 m χ is clearly visible in Fig. 2. The approximate expressionof the cross section close to the resonance region is (cid:104) σv (cid:105) ∼ m χ π (cid:104) (∆ m ) f + m χ π (cid:105) , (3.13)where ∆ m = 4 m χ − m σ , measuring the deviation from the exact location of the resonance.In this region (but above the blank region corresponding to Eq. (3.12)), a large value of f is required to reduce the otherwise too high annihilation cross section. Note, that once m χ ∼
40 TeV the cross section falls below the observed value even without a contributionfrom the resonance. Above those masses one does not expect any more resonant behavior,which is indeed what is reflected in Fig. 2. We note that numerical resolution affects the sizeof f that is displayed in Fig. 2 exactly on the resonance line, as f → ∞ for ∆ m →
0. Ofcourse, living exactly on resonance corresponds to an extremely fine-tuned parametric set-up. Note that beyond the mere parametric fine-tuning, another issue here is that f (cid:29) m σ would imply dynamical fine-tuning as well.We conclude the discussion of the scalar DM case by considering the scenario proposedin Ref. [12], that entertained the possibility of having the newly discovered Higgs-likeparticle itself be the dilaton. For the dilaton to mimic the Higgs, we must have m σ ≈ m h =126 GeV and f ≈ v = 246 . m σ and f , we find that the darkmatter mass that is needed for correct relic abundance is m χ ≈
52 GeV if the dark matteris a scalar. The leading annihilation channels at this value of m χ are to bottom and charmquarks and tau leptons. Larger values of m χ result in relic abundance that is too low,while lower values of m χ give a too-high relic abundance. This means that m χ ≈
52 GeVis an upper bound for scalar dark matter mass in our framework in the Higgs-like dilatonscenario. As we show in Sec. 4, such a low scalar dark matter mass is excluded by directdetection limits. Similar results are obtained for the case of fermion and vector DM, aspresented in the second and third plots in Fig. 2. The higgs-like dilaton scenario wouldrequire fermion dark matter of 61 GeV, or vector DM of 56 GeV. As we will see both ofthese cases are excluded by the direct detection bounds.Finally, note that in part of the parameter space depicted in Fig. 2 the DM annihila-tion cross section receives large non-perturbative corrections at low center of mass velocities(Sommerfeld enhancement). In our model, at large DM mass when the effective coupling m χ /f is not far from the perturbativity limit, the effect induces a sizable correction tothe relic abundance calculation. We compute the Sommerfeld enhancement in Sec. 5 andinclude it in a simplified form in the calculation of Fig. 2, by rescaling the tree-level an-nihilation cross section by the Sommerfeld enhancement factor at relative DM velocity v = 0 .
3, corresponding roughly to the thermal freeze-out kinematics. In most of the pa-rameter space, corresponding to perturbative coupling ( m χ /f ) / π (cid:28)
1, the correction to11he derived value of f ( m χ , m σ ) fixed by the relic abundance requirement is insignificant . Having defined the parameter space of the theory that reproduces the correct relic abun-dance, we now study direct detection constraints. For direct detection we need to considerthe elastic cross section of a dark matter particle that scatters off a nucleon. The dilatoninteracts with quarks q and the gluons G aµν inside a nucleon [20, 21]. Thus the relevantpart of the dilaton effective Lagrangian is L ⊃ − (cid:88) q σf (1 + γ q ) m q q ¯ q + α s πf c G G . (4.1)To estimate the anomalous dimension for quarks, one can consider the correspondingwarped extra dimensional models where the anomalous dimension is determined [4] by1 + γ = c L − c R , where c L,R are the bulk fermion mass parameters. For typical warpedfermion scenarios we find for example γ s ∼ .
16, which we neglect in the bounds below.Taking the matrix element between nucleon states yields the effective nucleon-dilatonLagrangian L σnn = y n σn ¯ n (4.2)where the coefficient y n is determined by the f nq , R n hadronic matrix elements: y n ≡ − (cid:88) q f nq m n f + R n c G πf . (4.3)For these matrix elements we use the values from [21–23]: f nq = (cid:104) n | ¯ qq | n (cid:105) m q m n f nu (cid:39) f nd (cid:39) . f ns (cid:39) . f nc (cid:39) . f nb (cid:39) . f nt (cid:39) . R n = α s (cid:104) n | G aµν G aµν | n (cid:105) (cid:39) − . In fine-tuned regions of the parameter space, where the Sommerfeld effect hits a resonance, DM anni-hilation re-coupling can significantly affect the relic abundance calculation [19]. We ignore this effect hereand comment about it in Sec. 5. σ χ,n ≈ y n π (cid:18) m χ f (cid:19) m n m σ (4.5)for either scalar, fermionic or vector dark matter.Fixing the scale f for given m σ and m χ to match the relic abundance, we plot theDM-nucleon elastic scattering cross section as a function of the dark matter mass for afew dilaton mass values. The results are illustrated on Fig. 3 along with the recent directdetection constraints from the LUX experiment [7]. We have also included the effects of thecollider bounds on the dilaton from the LHC and other machines (see Appendix A). Theseplots show that most of the parameter space is currently allowed both by the dark matterdirect detection experiments and also by the collider constraints, as long as m σ (cid:38)
200 GeV.As discussed in Sec. 3.3, for m χ (cid:29) m t and away from the resonance the annihilationcross section is proportional to m χ /f . Moreover, since y n ∝ /f , we can see that theelastic scattering cross section is proportional to the same combination m χ /f . Thus inthe appropriate regime the elastic cross section will be independent of the dark mattermass, as can be seen in Fig. 3. We now consider the prospects for indirect detection of dark matter annihilation via gammaray and cosmic ray antiproton flux measurements . We limit the discussion to the case inwhich the DM χ is a real scalar field. We expect similar results for the vector DM case; thefermion DM case will not have significant cosmic ray signatures as its annihilation is p-wavesuppressed in the small virial velocity of the Milky Way and its dwarf satellite galaxies.The parameter space of interest for the model includes the regime where m χ > m σ .In this regime, dilaton exchange produces an attractive Yukawa potential − αr e − m σ r , with α = m χ πf , that affects the dark matter annihilation process giving rise to Sommerfeldenhancement (SE; see e.g. [25, 26]) that needs to be taken into account in the indirectdetection estimates. In the top panel of Fig. 4 we plot the effective SE factor (denoted SE eff ) in the { m σ , m χ } plane, fixing the value of the scale f at each point to match theobserved dark matter relic abundance. We define SE eff as the value of the SE today in Additional constraints can be derived from neutrino experiments. These constraints are typicallyweaker than those arising from gamma ray and antiproton data (see e.g. [11] for discussion of the ν fluxin the context of a related model) and we do not consider them in this work. Under specific cosmicray propagation model assumptions, constraints can also be derived from the high energy positron flux.In comparison to the ¯ p calculation, however, the theoretical uncertainties for e + are larger as the resultsdepend crucially on the cosmic ray propagation time in the Galaxy that dictates the amount of e + radiativeenergy loss [24], and so we do not consider e + constraints in this work. v ∼ .
3. In ourcalculation we use an approximate formula for the SE factor [19, 27, 28], SE ≈ π(cid:15) v sinh (cid:16) (cid:15) v π(cid:15) φ (cid:17) cosh (cid:16) (cid:15) v π(cid:15) φ (cid:17) − cos (cid:34) π (cid:114) π (cid:15) φ − (cid:16) (cid:15) v π(cid:15) φ (cid:17) (cid:35) , (5.1)where (cid:15) v ≡ v α = πvf m χ and (cid:15) φ ≡ m σ αm χ = πm σ f m χ . We set the value of the dark matterparticles’ relative velocity to v = 10 − , appropriate for annihilation in the Galactic halo.We have verified that the approximation above reproduces the full Sommerfeld calculationto a good accuracy.The top panel of Fig. 4 shows that for DM mass above a few TeV, large values of theSE factor are possible with SE eff > in resonance regions. As we show below, this resultmay have interesting implications – striking indirect detection signatures are possible if themodel happens to live at an SE resonance. However, resonant SE is limited to fine-tunedregions in the parameter space. To illustrate this point, in the bottom panel we plot thevalue of SE vs. the DM mass fixing m σ = 3 TeV (corresponding to a vertical slice throughthe center of the top panel, marked by an arrow). For generic parameter configuration theeffective SE factor is modest, and only grows above 10 near resonances and for extremelyheavy DM mass, close to the unitarity limit where our calculation breaks down. Note thatwe truncate the value of SE eff at 10 in resonance peaks. As the resonance regions arefine-tuned, this has limited impact on our analysis. According to the analysis of [19], therelic abundance is depleted at the tip of these SE resonances due to chemical re-coupling ofDM at low redshifts, an effect that we do not include here and that would reduce the valueof SE eff . In addition, the low velocity divergence of the SE at the resonance tip should beregulated by bound-state decay that would also suppress the peak SE.In Secs. 5.1 and 5.2 below we calculate antiproton and gamma ray constraints onthe model. For antiprotons we adopt a conservative model-independent approach to theproblem of cosmic ray propagation, and provide some extra details to explain our method.The summary of our results is that the bulk of the parameter space of Fig. 4 (or equivalentlyFig. 2) is allowed by current constraints. This is not a surprise: much of the parameterspace consistent with the DM relic density corresponds to rather heavy m χ at the severalTeV, where current indirect searches do not yet constrain the thermal relic cross section.Indirect detection constraints do exclude, or make promising predictions for, the near-resonant SE regions seen in Fig. 4. If one accepts the assumption of a cusp DM densityprofile in the Milky Way Galactic Center, for example, then HESS gamma ray data alreadyexcludes much of the parameter region in the upper-left corner of the top panel of Fig. 4.15 ������������ FERMI DSph γ HESS GC γ [ m χ / GeV ] [ SE eff ] Figure 4: Top panel: Sommerfeld enhancement factor (SE) in the { m σ , m χ } plane. Abovethe dashed line m σ < f /
10, indicating fine-tuning. Bottom panel: SE vs. dark mattermass, fixing the dilaton mass to m σ = 3 TeV (marked on top panel with an arrow). Theregion above the red and green dashed lines is excluded by FERMI and HESS gamma rayobservations (the latter depend strongly on assumptions regarding the DM distributionin the Galaxy; see Sec. 5.2). The dark matter particles’ relative velocity today is set to v = 10 − . 16 .1 Antiprotons The PAMELA satellite experiment reported a measurement of the high energy antiprotonflux in interstellar space, extending up to 350 GeV [29]. The PAMELA measurementis consistent with model-independent calculations of the antiproton flux expected due tofragmentation of high energy primary cosmic ray nuclei on ambient interstellar gas in theGalaxy [30].Following Ref. [11], we derive a bound on the antiproton production in dark matterannihilation by imposing that the dark matter annihilation source of antiprotons in thelocal Galactic gas disc does not exceed the source due to the astrophysical production, inthe energy range covered by the current measurements. The bound derived in this manneris independent of modeling assumptions regarding the propagation of charged cosmic raysin the Galaxy. The bound is conservative because it does not include the possible additionalcontribution of DM annihilation in the cosmic ray halo that may extend well above andbelow the gas disc.The injection rate density of antiprotons due to DM annihilation is given by Q ¯ p,DM ( E ) = 12 n χ (cid:104) σv (cid:105) dN ¯ p dE ≈ × − cm − s − GeV − × (cid:16) ρ χ . − (cid:17) (cid:18) (cid:104) σv (cid:105) × − cm s − (cid:19) (cid:16) m χ (cid:17) − (cid:18) m χ dN ¯ p dE (cid:19) . (5.2)Here, ρ χ = m χ n χ ≈ . − is the DM mass density in the local halo and dN ¯ p dE is thedifferential antiproton spectrum per annihilation event. To compute dN ¯ p dE we use the codeprovided in Ref. [31], that directly produces the differential ¯ p spectrum for the channels χχ → W W, ZZ, hh, t ¯ t accounting for the decay and hadronization of the intermediateunstable states. To include the contribution of χχ → σσ we proceed in two steps. Firstwe use Ref. [31] to calculate the ¯ p spectrum arising in the dilaton rest frame due to dilatondecay; define this spectrum by (cid:104) dN ¯ p dE ( E ) (cid:105) σ → ¯ pX . We then convolve the dilaton decay ¯ p spectrum with the isotropic boost factor of the σ in the DM annihilation center of massframe, obtaining (cid:20) dN ¯ p dE ( E ) (cid:21) χχ → σσ = 1 γ σ β σ (cid:90) β − σ +1 β − σ − dxx (cid:20) dN ¯ p dE (cid:18) Exγ σ β σ (cid:19)(cid:21) σ → ¯ pX (5.3)where γ σ = m χ /m σ and β σ = (cid:112) − γ − σ . We neglect DM annihilation into gluons, since thebranching fraction of annihilation to this state is small compared to that of annihilation toquarks and massive gauge bosons. In the left panel of Fig. 5 we plot the differential flux of¯ p from DM annihilation with m χ = 6 . m σ = 427 GeV, and f = 6 .
00 200 500 1000 2000 5000 E (cid:64)
GeV (cid:68) p dE ΧΧ (cid:174) Γ X (cid:72) full (cid:76) ΧΧ (cid:174) bb (cid:174) Γ X
100 200 500 1000 2000 5000 E (cid:64)
GeV (cid:68) Γ dE Figure 5: Left: differential ¯ p spectrum per DM annihilation, computed for m χ = 6 . m σ = 427 GeV, and f = 6 . χχ → W W, ZZ, tt, σσ, ... ), while the blue line shows the spectrum due to χχ → bb alone.the disc is [30] Q ¯ p,CR ( E ) ≈ . × − cm − s − GeV − × (cid:18) E
100 GeV (cid:19) − . (cid:20) − .
22 log (cid:18) E
500 GeV (cid:19)(cid:21) J p (1 TeV) J p, (1 TeV) , (5.4)where J p (1 TeV) is the local proton flux sampled at E = 1 TeV and scaled to the measuredvalue J p, (1 TeV) ≈ × − GeV − cm − s − sr − . The uncertainties in the derivation ofEq. (5.4) are at the ∼
50% level. Our conservative bound on the DM annihilation rateamounts to imposing that the ratio Q ¯ p,CR ( E ) /Q ¯ p,DM ( E ) is larger than unity for E in therange 10-300 GeV.The basic result we find is that the model survives our antiproton constraint by alarge margin, unless it lives right on top of an SE resonance. If the model is near an SEresonance, then a detectable rise in the antiproton flux at high energy is predicted. For DMmass below about ∼
10 TeV, the rise would be in tension with currently available ¯ p dataand the model is observationally disfavored (again, only the region near an SE resonance,as seen in Fig. 4). For m χ (cid:38)
10 TeV, though, the rise in the ¯ p flux sets in at high energywith only a moderate effect in the energy range where current data exists. In this case,improved high energy cosmic ray measurements expected in the near future [32] may detectthe model in the ¯ p flux.We illustrate these findings in Fig. 6 where we plot the expected antiproton flux in ourmodel near an SE resonance for two chosen points. The data points (last one being an upperbound) and the green curve denote the PAMELA data and the secondary astrophysicsprediction, respectively. The red and magenta curves give an estimate of the antiprotonflux that would occur for the parameter points { m χ = 6 . , m σ = 300 GeV } and18 −4 −3 kinetic energy [GeV] pba r / p PAMELAsecondary prediction
Figure 6: Antiproton flux with DM annihilation at a Sommerfeld factor resonance. Datapoints and green curve denote PAMELA data and secondary astrophysics prediction, re-spectively. Red and magenta curves give a lower estimate of the ¯ p flux with DM an-nihilation for the model parameter point with { m χ = 6 . , m σ = 300 GeV } and { m χ = 31 TeV , m σ = 4 . } , respectively, where the SE factor is SE eff ≈ . { m χ = 31 TeV , m σ = 4 . } , respectively, where the effective SE factor is SE eff ≈ (fixing f to obtain the observed DM relic abundance).Above we chose tuned points with large SE eff to illustrate the possible ¯ p signal; asmentioned earlier, this large SE near the resonance peak can be damped somewhat by amore careful treatment of the relic abundance. However, we stress that the DM-inducedsignal depends on unknown cosmic ray propagation features. The red and magenta curvesin Fig. 6 should be considered as a robust lower bound on the DM-induced flux. Consideringdisc+halo diffusion models [33], for example, the actual flux could be as high as a factorof ∼
100 above the result we show here . A future detection of the model through cosmicray ¯ p is therefore conceivable also away from SE resonance peaks. The FERMI gamma ray telescope reported limits on DM annihilation based on a stackinganalysis of dwarf spheroidal galaxies [34]. The analysis is relatively insensitive to theassumed DM mass distribution in the target galaxies. Ref. [34] reports limits directly See App. B of Ref. [11] for a detailed discussion.
19n the annihilation cross section for the specific channel χχ → b ¯ b as a function of theDM mass. Using the code of Ref. [31] and following a similar method as that describedabove for the ¯ p spectrum calculation, we verified that the spectrum of continuum gammarays obtained in our model agrees to within a factor of 2-3 with the gamma ray spectrumresulting from a pure χχ → b ¯ b channel. In what follows we therefore assume that theconstraints quoted in [34] apply to our model directly. In the right panel of Fig. 5 we plotthe differential gamma ray flux from DM annihilation with m χ = 6 . m σ = 427 GeV,and f = 6 . χχ → W W, ZZ, tt, σσ, ... ), while the blue line shows the spectrum due to χχ → bb alone.We extrapolate the bound to m χ = 100 TeV, using the scaling m − χ dN γ dE ∼ m − χ thatapplies for photon energies in the FERMI range, E (cid:46)
500 GeV (cid:28) m χ . The resulting boundis illustrated by the red dashed line in the bottom panel of Fig. 4, focusing on a slice in theparameter space with m σ = 3 TeV.Stronger, but more model-dependent limits are obtained from ground-based air-Che-renkov telescopes. The HESS gamma ray observatory reported limits on DM annihilationbased on Galactic Center observations [35]. Due to the background subtraction methodof the experiment, the analysis is not sensitive to shallow DM density profiles, and so theresults are only applicable under the assumption of a cusp profile such as the Navaro-Frenk-White [36] distribution. Assuming a cusp distribution, neglecting the O (1) spectraldifference between the χχ → q ¯ q -induced gamma ray spectrum assumed in [35] and theactual spectrum in our model, and extrapolating their limits from m χ = 10 TeV up to m χ = 100 TeV, we obtain the bound depicted by the green dashed line in the bottom panelof Fig. 4.Finally, both FERMI [37] and HESS [38] reported limits on DM annihilation to agamma ray line. We calculate the branching fraction (cid:104) σv (cid:105) ( χχ → γγ ) / (cid:104) σv (cid:105) (total) usingEq. (B.3). This branching fraction is very small in our model, reminiscent of the resultfor a heavy Higgs. Consequently the gamma ray line constraint is sub-dominant comparedto the continuum emission bounds. We comment that the HESS limit [38] have recentlybeen used to put significant pressure on supersymmetric Wino dark matter [39, 40]. Thissituation is not reproduced here; for the Wino example, the strong exclusion is primarilydue to the presence of an electromagnetically charged state that is mass-degenerate withthe neutral DM particle, amplifying the di-photon annihilation diagram. Without a spe-cial construction of this kind, our dilaton-mediated DM scenario passes the line searchesunscathed. In this paper we explored the possibility that the dilaton could mediate dark matter anni-hilation. Such models have the appeal that the couplings are largely determined by scale20nvariance. The breaking scale of scale invariance f is fixed by requiring that the relic abun-dance matches the observed value, leaving the dark matter and dilaton masses as the mainfree parameters. We mapped the relevant { f, m χ , m σ } parameter space taking the variousdark matter annihilation modes into account and imposing unitarity bounds. We showedthat large regions of parameter space, with f, m χ , m σ all in the ∼ −
10 TeV range, cancorrectly reproduce the observed relic abundance. We find an upper bound f ≤ − m σ,χ .Collider searches for Higgs-like particles, including LHC, Tevatron and LEP analyses,put model dependent lower bounds on f for dilaton masses up to ∼ m χ (cid:46)
200 GeV. Current direct detectionexperiments yield similar model dependent exclusions for the lower end of the mass spec-trum, requiring m χ (cid:38)
300 GeV for m σ (cid:46)
300 GeV. The predicted dark matter-nucleonelastic scattering cross section becomes independent of the dark matter mass for heavydark matter.Our analysis of indirect detection included antiproton and gamma ray data and showsthat the bulk of the parameter space is consistent with the current constraints. A possiblesignal in high energy cosmic ray antiprotons could appear for favorable cosmic ray propaga-tion scenario for models with parameters close to a Sommerfeld enhancement resonance. Apromising avenue for probing the model all the way to very high DM mass is in high energyground-based gamma ray measurements, see e.g. [41, 42] for recent reviews. For scalar orvector DM, future gamma ray experiments should detect or exclude the entire parameterspace of the model.
Acknowledgements
We thank Andre Walker-Loud for pointing out the latest lattice QCD hadronic matrixelements. C.C. thanks the Aspen Center for Physics for its hospitality while part of thiswork was completed. S.L. thanks the particle theory group at Cornell University, theInstitute for Advanced Study in Princeton and the Mainz Institute for Theoretical Physics(MITP) for their hospitality and support while part of this work was completed. K.B. issupported by the DOE grant DE-SC000998. M.C. and C.C. are supported in part by theNSF grant PHY-1316222. SL is supported in part by the National Research Foundation ofKorea grant MEST No. 2012R1A2A2A01045722.
Note Added
While completing this project we became aware of [43] which investigates similar issues.21
Collider bounds
LHC
LHCTEVLEP
LHC model Amodel B m σ [ GeV ] f [ G e V ] Figure 7: 95% C.L. collider exclusion limit on the scale of conformal symmetry breaking, f , with respect to m σ for our benchmark models A and B.As mentioned in the main text, in addition to the direct detection bounds there arealso collider bounds from the LHC and earlier experiments. The dilaton (roughly) mimicsa Higgs boson, with couplings to massive SM fields suppressed by the factor v/f comparedto that of the Higgs and couplings to massless gauge bosons that involve contributions fromthe matter content of the conformal sector. Collider bounds on the dilaton can thus beobtained by recasting the results of direct production limits from Higgs boson searches. Weuse the HiggsBound [44–46] code version 4.1.2, that incorporates all the currently availableexperimental analyses from LEP, the Tevatron, and the LHC [44–46].The resulting collider bounds on the conformal symmetry breaking scale f as a functionof the dilaton mass is presented in Fig. 7 for the two benchmark models A and B definedin Sec. 2. In obtaining these bounds we assumed, for simplicity, no invisible decay channelsfor the dilaton. We can see that the collider bounds are strongly model dependent: modelA has a large coupling to gluons, and thus is very strongly constrained throughout theparameter space relevant for LHC kinematics. Model B has small couplings to gluons andphotons, and is only weakly constrained for dilaton masses above 200 GeV.The resulting bound on f can be turned into a bound on m χ using Fig. 2. For examplethe f (cid:38) m σ (cid:46)
400 GeV in model A implies m χ (cid:38)
300 GeV, with theexception for a narrow resonance region. 22
Additional annihilation channels
B.1 Scalar dark matter
Annihilation to fermions of mass m ψ : σv ( χχ → ¯ ψψ ) (cid:39) m ψ m χ (cid:0) m χ − m ψ (cid:1) / πf | m χ − m σ − i Im (cid:0) Π(4 m χ ) (cid:1) | (B.1)Annihilation to a real scalar of mass m h : σv ( χχ → hh ) (cid:39) m χ m h (cid:113) m χ − m h πf | m χ − m σ − i Im (cid:0) Π(4 m χ ) (cid:1) | (B.2)While annihilation to photons is negligible for the relic abundance calculation, it is impor-tant for indirect detection. We get σv ( χχ → γγ ) (cid:39) m χ α c π f | m χ − m σ − i Im (cid:0) Π(4 m χ ) (cid:1) | (B.3)where c EM encodes the coupling of photons to dilaton: c EM = (cid:32) F W ( x W ) − (cid:88) f N c Q f F f ( x f ) + b (EM)IR − b (EM)UV (cid:33) (B.4) x i = m i m χ (B.5) F W ( x ) = 2 + 3 x + 3 x (2 − x ) f ( x ) (B.6) F f ( x ) = 2 x [1 + (1 − x ) f ( x )] (B.7) f ( x ) = (cid:40) arcsin(1 / √ x ) : x ≥ − (cid:104) log (cid:16) √ x − −√ x − (cid:17) − iπ (cid:105) : x < B.2 Fermion dark matter
Annihilation to fermions of mass m ψ : σv ( ¯ χχ → ¯ ψψ ) (cid:39) v m ψ m χ (cid:0) m χ − m ψ (cid:1) / πf | m χ − m σ − i Im (cid:0) Π(4 m χ ) (cid:1) | (B.9)Annihilation to a real scalar of mass m h : σv ( ¯ χχ → hh ) (cid:39) v m χ m h (cid:113) m χ − m h πf | m χ − m σ − i Im (cid:0) Π(4 m χ ) (cid:1) | (B.10)23 .3 Vector dark matter Annihilation to fermions of mass m ψ : σv ( χχ → ¯ ψψ ) (cid:39) m ψ m χ (cid:0) m χ − m ψ (cid:1) / πf | m χ − m σ − i Im (cid:0) Π(4 m χ ) (cid:1) | (B.11)Annihilation to a real scalar of mass m h : σv ( χχ → hh ) (cid:39) m χ m h (cid:113) m χ − m h πf | m χ − m σ − i Im (cid:0) Π(4 m χ ) (cid:1) | (B.12) C Dilaton decay channels
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